Given categories $X$ and $Y$ and a strong functor

$$D:X^{op}\times Y\to Cat$$

we can of course build the oplax colimit

$$\mathrm{colim}^{oplax}_{X^{op}\times Y}D$$

via the usual (covariant) grothendieck construction:

*Objects* are triples $(x,d,y)$ with $d\in D(x,y)$.

*Morphisms* $(x,d,y)\to (x',d',y')$ are triples $(f,\phi, g)$ with $f:x'\to x$, $g:y\to y'$ and $$\varphi: f^*x_*g\to x'.$$

There is however another possible construction that corresponds better to the slogan "presheaves are distributors into the one-point category and copresheaves are distributors out of the one-point category":

*Objects* are the same triples as above.

*Morphisms* however are now triples $(f,\varphi, g)$ with $f:x\to x'$ (the direction changed!) and $g:y\to y'$ and

$$\varphi:x_*g \to f^*x'.$$

Why does it correspond better to the slogan stated above? Taking distributors having at one side the one-point category then specialises to the usual grothendieck constructions yielding fibered and opfibered categories respectively.

**Question:** What can be said about the second construction? Is it some kind of colimit as well?